Related questions with answers

Reliability of a manufacturing network. A team of industrial management university professors investigated the reliability of a manufacturing system that involves multiple production lines (Journal of Systems Sciences & Systems Engineering, March 2013). An example of such a network is a system for producing integrated circuit (IC) cards with two production lines set up in sequence. Items (IC cards) first pass through Line 1 , then are processed by Line 2. The probability distribution of the maximum capacity level $(x)$ of each line is shown below. Assume the lines operate independently. Find the mean maximum capacity for each line. Interpret the results practically.

Line	Maximum Capacity, $x$	$p(x)$
1	0	$.01$
	12	$.02$
	24	$.02$
36	$.95$

2	0	$.002$
	35	$.002$
	70	$.996$

A tea.in of industrial management university professors investigated the reliability of a manufacturing system that involves multiple production lines (Journal of Systems Sciences & Systems Engineering, Mar. 2013). An example of such a network is a system for producing integrated circuit (IC) cards with two production lines set up in sequence. Items (IC cards) first pass through Line 1 and then are processed by Line 2. Toe probability distribution of the maximum capacity level of each line is shown in the next column. Assume the lines operate independently.

\begin{matrix} \text{Line} & \text{Maximum Capacity, x} & \text{p(x)}\\ \text{1} & \text{0} & \text{.01}\\ \text{1} & \text{12} & \text{.02}\\ \text{1} & \text{24} & \text{.02}\\ \text{1} & \text{36} & \text{.95}\\ \text{2} & \text{0} & \text{.002}\\ \text{2} & \text{35} & \text{.002}\\ \text{2} & \text{70} & \text{.996}\\ \end{matrix}

Now consider the network of two production lines. What is the probability that a maximum capacity level of 30 items is maintained throughout the network?

\begin{matrix} \text{Line} & \text{Maximum Capacity, x} & \text{p(x)}\\ \text{1} & \text{0} & \text{.01}\\ \text{1} & \text{12} & \text{.02}\\ \text{1} & \text{24} & \text{.02}\\ \text{1} & \text{36} & \text{.95}\\ \text{2} & \text{0} & \text{.002}\\ \text{2} & \text{35} & \text{.002}\\ \text{2} & \text{70} & \text{.996}\\ \end{matrix}

Find the probability that the maximum capacity level for Line 1 will exceed 30 items.

Question

Reliability of a manufacturing network. A team of industrial management university professors investigated the reliability of a manufacturing system that involves multiple production lines (Journal of Systems Sciences & Systems Engineering, March 2013). An example of such a network is a system for producing integrated circuit (IC) cards with two production lines set up in sequence. Items (IC cards) first pass through Line 1 , then are processed by Line 2. The probability distribution of the maximum capacity level $(x)$ of each line is shown below. Assume the lines operate independently. Find the standard deviation of the maximum capacity for each line. Then, give an interval for each line that will contain the maximum capacity with probability of at least .75.

Line Maximum Capacity, $x$ $p(x)$

1 0 $.01$

12 $.02$

24 $.02$

36 $.95$

2 0 $.002$

35 $.002$

70 $.996$

Solution

Verified

Answered 1 year ago

Step 1

1 of 8

We need to determine the standard deviation and determine an interval containing the maximum capacity with at least a 25% chance.

Chebyshev's Rule tells us that:

At least none of the data values are within one standard deviation of the mean.
At least 3/4 or 75% of the data values are within two standard deviations of the mean.
At least 8/9 or about 89% of the data values are within three standard deviations of the mean.

Thus the interval of interest will be two standard deviations about the mean.